JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (8): 75-80.doi: 10.6040/j.issn.1671-9352.0.2016.458

Previous Articles     Next Articles

Relative torsionless modules

XU Hui1,2, ZHAO Zhi-bing1*   

  1. 1. School of Mathematical Scicences, Anhui University, Hefei 230601, Anhui, China;
    2. Department of Basic Teaching, Anhui Business Vocational College, Hefei 231131, Anhui, China
  • Received:2016-09-22 Online:2017-08-20 Published:2017-08-03

PDF (PC)

576

Abstract: Some elementary properties of relative torsionless modules with respect to a faithful and balanced bimodule are studied. Some equivalent conditions for a left or right Noether ring is a QF-ring is gotten. It is proved that a finitely generated module is an ω-torsionless module if and only if it is an ω-1-syzygy module if and only if it is an ω-1-torsionfree module when ω is a generalized tilting bimodule. Moreover, the extension closure of the class of ω-torsionless modules is investigated, and some classical results are generalized to the relative case.

Key words: extension closure, ω-torsionless modules, ω-reexive modules, (strong)grade of modules related to ω

CLC Number: 

  • O154.2
[1] ANDERSON F M, FULLER K R. Rings and categories of modules[M] // GTM13. New York: Springer-Verlag, 1992.
[2] LUO Rong, HUANG Zhaoyong. When are torsionless modules projective[J]. J Algebra, 2007, 320:2156-2164.
[3] SAZEEDEH R. Strongly torsionfree, copure flat and Matlis reexive modules[J]. J Pure Appl Algebra, 2004, 192(3):265-274.
[4] SALIMI M, TAVASOLI E, YASSEMI S. k-torsionless modules with finite Gorenstein dimension[J]. Czechoslovak Mathematical Journal, 2012, 62(3):663-672.
[5] XIN Lin, WEI Chunjin. On U-torsionless modules and U-neat modules[J]. East-West J Math, 2000, 2(1):41-48.
[6] HUANG Zhaoyong. ω-k-torsionfree modules and ω-left approximation dimension[J]. Sci China(Series A), 2000, 44:184-192.
[7] HUANG Zhaoyong. Extension closure of relative syzygy modules[J]. Sci China(Series A), 2003,4 6(5): 611-620.
[8] AUSLANDER M, REITEN I. Sygyzy modules for noetherian rings[J]. J Algebra, 1996, 183:167-185.
[1] CHEN Wen-qian, ZHANG Xiao-jin, ZAN Li-bo. The number of tilting modules over Gorenstein algebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 14-16.
[2] GUO Shou-tao, WANG Zhan-ping. Gorenstein homological dimensions of modules under exact zero-divisors [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 17-21.
[3] . FR-injective and FR-flat dimensions of complexes [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 1-6.
[4] YANG Chun-hua. A note on the Gc-injective dimension of a complex [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(11): 82-86.
[5] LUO Xiao-qiang, XING Jian-min. Ding gr-injective and gr-flat modules [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(11): 87-91.
[6] WANG Xiao-qing, LIANG Li. Strongly cotorsion modules under faithfully flat co-base change [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(11): 92-94.
[7] CHEN Xiu-li, CHEN Jian-long. Homological dimensions with respect to semidualizing modules and excellent extensions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 85-89.
[8] SUN Wei-kun, LIN Han-xing. Representation dimension of one-point extension algebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(6): 85-91.
[9] XIE Zong-zhen, ZHANG Xiao-jin. On algebras with all τ-rigid modules projective [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 16-20.
[10] CHENG Hai-xia, YIN Xiao-bin. Gorenstein injective objects in Abelian categories [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 79-84.
[11] CHEN Xiu-li, CHEN Jian-long. Copure projective dimensions under Hopf extensions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(10): 7-10.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd